Sunday Times Teaser 2617 – A Case of Ambiguity

by Des MacHale

Triangles that have two of their sides of the same length and one of their angles the same are not necessarily congruent when the common angle is not that between the common sides.

I have found such a pair of non-congruent triangles in which all side lengths are integers and for which the length of the longest sides of both triangles is 10.

What are the lengths of the other two sides of the smaller triangle?